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Nomenclature
Roman letters
a
A
A
b
Ba
c
c
cκ
cp
cs
cv
C
C
CD
Cij
CL
Cp
Cpmin
d
dj
do
D
D
Dm
Ds
Amplitude of wave-like disturbance
Cross-sectional area or cloud radius
Attenuation
Power law index
Bagnold number, ρS D2 γ̇/µL
Concentration
Speed of sound
Phase velocity for wavenumber κ
Specific heat at constant pressure
Specific heat of solid or liquid
Specific heat at constant volume
Compliance
Damping coefficient
Drag coefficient
Drag and lift coefficient matrix
Lift coefficient
Coefficient of pressure
Minimum coefficient of pressure
Diameter
Jet diameter
Hopper opening diameter
Particle, droplet or bubble diameter
Mass diffusivity
Volume (or mass) mean diameter
Sauter mean diameter
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D(T )
D
e
E
f
f
fL , fV
Fi
Fr
F
g
gL , gV
GN i
GN
h
h
H
H
He
Hm
i, j, k, m, n
i
I
I
ji
jN i
jN
k
k
k
kL , kV
K
K∗
Kc
Kij
Kn , Ks
Kn
K
Determinant of the transfer matrix [T ]
Thermal diffusivity
Specific internal energy
Rate of exchange of energy per unit volume
Frequency in Hz
Friction factor
Liquid and vapor thermodynamic quantities
Force vector
Froude number
Interactive force per unit volume
Acceleration due to gravity
Liquid and vapor thermodynamic quantities
Mass flux of component N in direction i
Mass flux of component N
Specific enthalpy
Height
Height
Total head, pT /ρg
Henry’s law constant
Haberman-Morton number, normally gµ4 /ρS 3
Indices
Square root of −1
Acoustic impulse
Rate of transfer of mass per unit volume
Total volumetric flux in direction i
Volumetric flux of component N in direction i
Volumetric flux of component N
Polytropic constant
Thermal conductivity
Boltzmann’s constant
Liquid and vapor quantities
Constant
Cavitation compliance
Keulegan-Carpenter number
Added mass coefficient matrix
Elastic spring constants in normal and tangential directions
Knudsen number, λ/2R
Frictional constants
Typical dimension
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t
L
L
m
ṁ
mG
mp
M
M∗
Mij
M
Ma
n
ṅ
ni
N (R), N (D), N(v)
N∗
Nu
p
pT
pa
pG
ps
P
Pe
Pr
q
qi
Q
Q
Q
r, ri
rd
R
R∗k
RB
Re
Re
R
Turbulent length scale
Inertance
Latent heat of vaporization
Mass
Mass flow rate
Mass of gas in bubble
Mass of particle
Mach number
Mass flow gain factor
Added mass matrix
Molecular weight
Martinelli parameter
Number of particles per unit volume
Number of events per unit time
Unit vector in the i direction
Particle size distribution functions
Number of sites per unit area
Nusselt number
Pressure
Total pressure
Radiated acoustic pressure
Partial pressure of gas
Sound pressure level
Perimeter
Peclet number, usually W R/αC
Prandtl number, ρνcp/k
General variable
Heat flux vector
General variable
Rate of heat transfer or release per unit mass
Rate of heat addition per unit length of pipe
Radial coordinate and position vector
Impeller discharge radius
Bubble, particle or droplet radius
Resistance of component, k
1
Equivalent volumetric radius, (3τ /4π) 3
Equilibrium radius
Reynolds number, usually 2W R/νC
Gas constant
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s
s
s
S
SD
St
Str
t
tc
tu
tT
T
T
Tij
ui
uN i
ur , uθ
us
u∗
U, Ui
U∞
v
V, Vi
V
V
V̇
w
W, Wi
W∞
Wp
Wt
We
W
x, y, z
xi
x
X
z
Coordinate measured along a streamline or pipe centerline
Laplace transform variable
Specific entropy
Surface tension
Surface of the disperse phase
Stokes number
Strouhal number
Time
Binary collision time
Relaxation time for particle velocity
Relaxation time for particle temperature
Temperature
Granular temperature
Transfer matrix
Velocity vector
Velocity of component N in direction i
Velocity components in polar coordinates
Shock velocity
Friction velocity
Fluid velocity and velocity vector in absence of particle
Velocity of upstream uniform flow
Volume of particle, droplet or bubble
Absolute velocity and velocity vector of particle
Volume
Control volume
Volume flow rate
Dimensionless relative velocity, W/W∞
Relative velocity of particle and relative velocity vector
Terminal velocity of particle
Typical phase separation velocity
Typical phase mixing velocity
Weber number, 2ρW 2 R/S
Rate of work done per unit mass
Cartesian coordinates
Position vector
Mass fraction
Mass quality
Coordinate measured vertically upward
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Greek letters
α
β
γ
γ̇
Γ
δ
δd
δm
δT
δ2
δij
ζ
η
θ
θ
θw
κ
κ
κL , κG
λ
λ
λ
Λ
µ
µ∗
ν
ν
ξ
ρ
σ
σi
σij
D
σij
Σ(T )
Volume fraction
Volume quality
Ratio of specific heats of gas
Shear rate
Rate of dissipation of energy per unit volume
Boundary layer thickness
Damping coefficient
Fractional mass
Thermal boundary layer thickness
Momentum thickness of the boundary layer
Kronecker delta: δij = 1 for i = j; δij = 0 for i = j
Fractional volume
Coefficient of restitution
Rate of dissipation of energy per unit mass
Attenuation or amplification rate
Bubble population per unit liquid volume
Angular coordinate or direction of velocity vector
Reduced frequency
Hopper opening half-angle
Wavenumber
Bulk modulus of compressibility
Shape constants
Wavelength
Mean free path
Kolmogorov length scale
Integral length scale of the turbulence
Dynamic viscosity
Coulomb friction coefficient
Kinematic viscosity
Mass-based stoichiometric coefficient
Particle loading
Density
Cavitation number
Inception cavitation number
Stress tensor
Deviatoric stress tensor
Thermodynamic parameter
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τ
τi
τn
τs
τw
ψ
ψ
φ
φ
φ
φ2L , φ2G , φ2L0
ϕ
ω
ωa
ωi
ωn
ωm
ωm
ωp
Ω
Kolmogorov time scale
Interfacial shear stress
Normal stress
Shear stress
Wall shear stress
Stokes stream function
Head coefficient, ∆pT /ρΩ2 rd2
Velocity potential
Internal friction angle
Flow coefficient, j/Ωrd
Martinelli pressure gradient ratios
Fractional perturbation in bubble radius
Radian frequency
Acoustic mode frequency
Instability frequency
Natural frequency
Cloud natural frequencies
Manometer frequency
Peak frequency
Rotating frequency (radians/sec)
Subscripts
On any variable, Q:
Qo
Q1 , Q2 , Q3
Q1 , Q2
Q∞
Q∗
QA
Qb
QB
QB
QC
Qc
QD
Initial value, upstream value or reservoir value
Components of Q in three Cartesian directions
Values upstream and downstream of a component or flow structure
Value far from the particle or bubble
Throat values
Pertaining to a general phase or component, A
Pertaining to the bulk
Pertaining to a general phase or component, B
Value in the bubble
Pertaining to the continuous phase or component, C
Critical values and values at the critical point
Pertaining to the disperse phase or component, D
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Qe
Qe
QG
Qi
Qij
QL
Qm
QN
QO
Qr
Qs
QS
QV
Qw
Qθ
Equilibrium value or value on the saturated liquid/vapor line
Effective value or exit value
Pertaining to the gas phase or component
Components of vector Q
Components of tensor Q
Pertaining to the liquid phase or component
Maximum value of Q
Pertaining to a general phase or component, N
Pertaining to the oxidant
Component in the r direction
A surface, system or shock value
Pertaining to the solid particles
Pertaining to the vapor phase or component
Value at the wall
Component in the θ direction
Superscripts and other qualifiers
On any variable, Q:
Q , Q, Q∗
Q̄
Q̀
Q̃
Q̇
Q̈
Q̂(s)
Q̆
δQ
Re{Q}
Im{Q}
Used to differentiate quantities similar to Q
Mean value of Q or complex conjugate of Q
Small perturbation in Q
Complex amplitude of oscillating Q
Time derivative of Q
Second time derivative of Q
Laplace transform of Q(t)
Coordinate with origin at image point
Small change in Q
Real part of Q
Imaginary part of Q
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NOTES
Notation
The reader is referred to section 1.1.3 for a more complete description of
the multiphase flow notation employed in this book. Note also that a few
symbols that are only used locally in the text have been omitted from the
above lists.
Units
In most of this book, the emphasis is placed on the nondimensional parameters that govern the phenomenon being discussed. However, there are
also circumstances in which we shall utilize dimensional thermodynamic and
transport properties. In such cases the International System of Units will be
employed using the basic units of mass (kg), length (m), time (s), and absolute temperature (K).
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